Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C52 ⋊ #1884 |
974 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,3)(2,7)(4,10)(5,11)(6,13)(8,15)(9,14)(12,16)(17,18)(19,20)(22,23)(24,25);(2,6,8,9)(7,14,15,13)(18,19)(20,21)(23,24)(25,26);(1,2,4,9,5,8,12,6)(3,7,10,14,11,15,16,13)(23,24)(25,26)Polycyclic presentation:
2^7,5^2|a^2=e,b^2=d,e^2=g,b^a=bef,c^b=cd,d^a=dg,f^a=fg,f^c=fg,h^a=h^4,h^b=h^4,h^c=h^4,i^a=h^3i,i^b=i^4,i^c=i^4200 conjugacy classes — view the full character table.
| maximal subgroup of | #72226431 |
| maximal normal subgroup of | #72226431 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 2, F.8^4);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^4);;
gap> SetConjugate(coll, 9, 3, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);